Lie symmetries in nonlinear liquid film flow with heat and mass transfer and variable magnetic field

This paper is focused on the study of coupled heat and mass transfer in a liquid film over an unsteady stretching surface. The stretching rate and temperature of the sheet vary with time. The governing partial differential equations are transformed into ordinary differential equations by using similarity transformations. The resulting similarity equations are solved analytically by an efficient perturbation expansion method. A parametric study illustrating the influence of the unsteadiness parameter and Prandtl number on the fluid velocity as well as temperature is conducted.

An analysis has been carried out to study the combined effects of nonlinear thermal radiation and internal heat source/sink on heat and mass transfer in a thin liquid film over a permeable unsteady stretching sheet with suction/injection in the presence of chemical reaction. Since the momentum, energy and mass diffusion equations are highly non-linear, the problem is solved numerically. The governing equations are first reduced to a set of ordinary differential equations by using the similarity transformations and then the resulting nonlinear ordinary differential equations are solved using Runge–Kutta–Fehlberg method with shooting technique. The effects of various physical parameters on heat and mass transfer in a thin liquid film are presented graphically and in tabular forms. Numerical results are compared with the published results for some limiting cases. It is found that increase in the unsteadiness parameter leads to increase in the velocity distribution, temperature gradient and concentration gradient due to reduction in the thin film thickness. Also, increase in the thermal radiation parameter decreases the temperature gradient whereas reverse effect is seen on the concentration gradient by increasing the values of the Schmidt number.

This paper describes nonlinear thermal radiation effects on MHD heat and mass transfer in a thin liquid film over a permeable unsteady stretching surface taking temperature-dependent fluid viscosity with convective boundary condition. For the non-linearity of the momentum, energy and mass diffusion equations, the problem is solved numerically. At first, Similarity transformations is used to the governing equations to reduce the equations into a set of ordinary differential equations. Then the resulting nonlinear ordinary differential equations are solved using Runge-Kutta-Felberg method with shooting technique. Different physical parameters effects on heat and mass transfer in a thin liquid film are presented graphically. It is found that increase in the unsteadiness parameter leads to increase in the velocity distribution, temperature and concentration gradient. Further, increase in the value of magnetic parameter results in a decrease in the velocity profile and increase in the temperature and concentration gradient. For enhancement of thermal radiation decreases the temperature gradient of the thin film flow. Also, for increase in viscosity variation parameter is to decrease velocity distribution but reverse effects shown in case of temperature and concentration gradient.

The Lie symmetry method is applied, and exact homotopic solutions of a non-linear double-diffusion problem are obtained. Additionally, we derived Lie point symmetries and corresponding transformations for equations representing heat and mass transfer in a thin liquid film over an unsteady stretching surface, using MAPLE. We used these symmetries to construct new (Lie) similarity transformations that are different from those that are commonly used for flow and mass transfer problems. These new (Lie) similarity transformations map the partial differential equations of a mathematical model under consideration to ordinary differential equations along with boundary conditions. Lie similarity transformations are shown to lead to new solutions for the considered flow problem. These solutions are obtained using the homotopy analysis method to analytically solve the ordinary differential equations that resulted from the reduction of considered flow equations through Lie similarity transformations. With the aid of these solutions, effects of various parameters on the flow and heat transfer are discussed and presented graphically in this study.

Numerous flow and heat transfer studies have relied on the construction of similarity transformations which map the nonlinear partial differential equations (PDEs) describing the flow and heat transfer, to ordinary differential equations (ODEs). For these reduced equations, one finds multiple analytic and approximate solution procedures as compared to the flow PDEs. Here, we aim at constructing multiple classes of similarity transformations that are different from those already existing in the literature. We adopt the Lie symmetry method to derive these new similarity transformations which reveal new classes of ODEs corresponding to flow equations when applied to them. With these multiple classes of similarity transformations, one finds multiple reductions in the flow PDEs to ODEs. On solving these ODEs analytically or numerically, we obtain different kinds of flow and heat transfer patterns that help in determining optimized solutions in accordance with the physical requirements of a problem. For the said purpose, we derive Lie point symmetries for the magnetohydrodynamic Casson fluid flow and heat transfer in a thin film on an unsteady stretching sheet with viscous dissipation. Linear combinations of these Lie symmetries that are again the Lie symmetries of the flow model are employed here to construct new similarity transformations. We derive multiple Lie similarity transformations through the proposed procedure which lead us to more than one class of reduced ODEs obtained by applying the deduced transformations. We analyze the flow and heat transfer by deriving analytic solutions for the obtained classes of systems of ODEs using the homotopy analysis method. Magnetic parameters and viscous dissipation influences on the flow and heat transports are investigated and presented in graphical and tabulated formats.

In thin films on unsteady stretching surfaces, flow and heat transfer under different physical conditions are studied to refine countless industrial products. In this work we consider such a flow with an external magnetic field and apply Lie symmetry method on it to analyze the flow and heat transfer. We construct Lie point symmetry generators for this magnetohydrodynamic flow and heat transfer in a thin liquid film over an unsteady stretching surface. We obtain a six dimensional Lie point symmetry algebra and by employing a few of these symmetry generators we deduce invariants and similarity transformations for the flow model. We employ these transformations on the flow equations that are partial differential equations to reduce their independent variables from three to one. This procedure reveals corresponding systems of ordinary differential equations. The similarity transformations obtained here are different from those that are derived earlier for the considered flow problem. Hence by employing these transformations on the flow model leads to new classes of ordinary differential equations. It implies existence of additional classes of Lie similarity transformations to study the MHD flow and heat transfer in a thin film on an unsteady stretching sheet. We obtain analytic solutions for these reduced systems of equations with the homotopy analysis method to present effects of the Prandtl number, magnetic and unsteadiness parameters on the velocity and temperature profiles. These are new solutions for the considered flow model as Lie symmetries have not been employed earlier on it. All results are presented with the help of graphs and tables.

The aim of this article is to investigate the effect of mass and heat transfer on unsteady squeeze flow of viscous fluid under the influence of variable magnetic field. The flow is observed in a rotating channel. The unsteady equations of mass and momentum conservation are coupled with the variable magnetic field and energy equations. By using some appropriate similarity transformations, the partial differential equations obtained are then converted into a system of ordinary differential equations and are solved by Homotopy Analysis Method (HAM). The influence of the natural parameters are investigated for the velocity field components, magnetic field components, heat and mass transfer. A direct effect of the squeeze Reynold number is observed on both concentration and temperature. Moreover, increasing the magnetic Reynold number shows an increase in the fluid temperature, but in the case of concentration, an inverse relation is observed. Furthermore, a decreasing effect of the Dufour number is observed on both concentration and temperature distribution. Besides, in case of the Soret number, a direct effect is observed on concentration, but an inverse effect can be seen on temperature distribution. Different effects are shown through graphs in this study and an error analysis is also presented through tables and graphs.

The effect of flow distribution on heat and mass transfer of MHD thin liquid film flow over an unsteady stretching sheet in the presence of variational physical properties with mixed convection

Heat transfer in MHD thin film flow with concentration using lie point symmetry approach

Encountering heat and mass transfer mechanisms simultaneously in Powell-Erying fluid through Lie symmetry approach

The heat and mass transfer characteristics of a liquid film which contain thermosolutal capillarity and a variable magnetic field over an unsteady stretching sheet have been investigated. The governing equations for momentum, energy, and concentration are established and transformed to a set of coupled ordinary equations with the aid of similarity transformation. The analytical solutions are obtained using the double-parameter transformation perturbation expansion method. The effects of various relevant parameters such as unsteady parameter, Prandtl number, Schmidt number, thermocapillary number, and solutal capillary number on the velocity, temperature, and concentration fields are discussed and presented graphically. Results show that increasing values of thermocapillary number and solutal capillary number both lead to a decrease in the temperature and concentration fields. Furthermore, the influences of thermocapillary number on various fields are more remarkable in comparison to the solutal capillary number.

The work is devoted to the study of heat and mass transfer in a liquid film flowing down on a heated surface under conditions of evaporation into a crossflow of a gas neutral with respect to the liquid. The work aimed to experimentally determine the average heat transfer coefficients from a heated surface to the film, heat transfer and mass transfer from the film to the gas flow and to establish their dependence on the input parameters of the heat and mass transfer process. To achieve this goal, an experimental setup was created, and a research technique was developed based on the proposed mathematical model of the heat and mass transfer process. The results of the study showed that the dependences of the average heat and mass transfer coefficients on the initial liquid flow rate are extreme with the minimum values of these coefficients at the liquid flow rate, which corresponds to the critical value of the Reynolds criterion Re l cr ≈ 500, which indicates a transition from the laminar falling films to turbulent mode under the considered conditions. The dependences of the heat and mass transfer coefficients on other process parameters for both modes of film falling are established. A generalization of the experimental data made it possible to obtain empirical equations for calculating these coefficients. Keywords: heat and mass transfer, cross flow, film apparatus, heat and mass return coefficient, neutral gas.

Coupled mass and heat transfer in a multicomponent turbulent falling liquid film

The current study explores the impact of heat source/sink on the flow of nanofluid across an exponentially stretchable sheet with the suspension of TiO2 as nanoparticle in base liquid water. Furthermore, activation energy, Newtonian heating and porous medium are accounted in the modelling. The modelling equations are transformed into a system of ordinary differential equations (ODEs) using similarity transformations. To solve these equations numerically, the Runge Kutta Fehlberg 45 (RKF 45) method and shooting approach are used. Graphical representations are used to study and address the effect of important factors on flow fields, heat, and mass transfer rates. The outcome reveals that, rising values of porous and suction parameters declines the velocity field. The rise in values of heat source/sink parameter declines the heat transfer. Moreover, rate of declination in heat transfer is faster for Newtonian heating (NH) case than common wall temperature (CWT) case. The rise in values of volume fraction and heat source/sink parameter declines the Nusselt number for both CWT and NH cases. Further, the fluid shows improved rate of heat transfer for CWT case than NH case.

This research aims to enhance industrial processes and address environmental issues by studying the effects of suction and injection on the Couette flow of a viscoelastic dusty fluid over a porous oscillating plate in a rotating frame. We investigate how these factors influence fluid flow, heat, mass, and temperature transfer rates. Suction and blowing are common techniques for controlling fluid flow in conduits. The study considers the impact of the right plate's oscillation on fluid velocity along the x‐axis, as well as mass diffusion and thermal effects from the heating of the right plate. Due to the rotation, the velocities of both the fluid and dust particles exhibit complex primary and secondary behaviors. Also considered the dust particles are homogenously distributed in the base fluid. Furthermore, the variable temperature and concentration are associated with boundary conditions. We develop a mathematical model using partial differential equations (PDEs) to describe these phenomena. The governing equations are non‐dimensionalized and transformed into ordinary differential equations (ODEs) using periodic solutions and solved via the Poincare‐Lighthill perturbation method. Key engineering parameters such as the Sherwood number, Nusselt number, and skin friction are calculated. The study reveals the influence of uniform suction/injection on velocity and temperature distributions, showing that suction can significantly alter boundary layer development, even under resonance conditions. The radiation parameter, Grashof number, and second‐grade parameter cause a decrease in skin friction as their values increase. On the other hand, suction, rotation, magnetic, dusty fluid, and Reynold numbers cause a rise in skin friction.The objective of this research is to improve industrial processes and tackle environmental issues by examining the effects of suction and injection on the Couette flow of a viscoelastic dusty fluid over a porous oscillation plate in a rotating frame. The purpose of this study is to determine how suction/injection influences the Couette flow of a viscoelastic dusty fluid, as well as the heat, mass, and temperature transfer rates across a porous, oscillating plate in a rotating frame. It also takes heat transfer into account. When manipulating the flow of a fluid via a conduit, suction and blowing are common methods. As the right plate oscillates, the x‐axis becomes the primary consideration for fluid velocity. We also take into account the mass diffusion and thermal influence on the flow from the right plate's heating. As a consequence of the spin, the fluid, and dust particles have velocities that are both secondary and primary, making them complicated. Using PDEs, we may translate the aforementioned physical phenomena into a mathematical model of the relevant flow regime. The system of governing equations is non‐dimensionalized by the use of appropriate nondimensional variables. The system of PDEs is transformed into a system of ODEs with the help of assumed periodic solutions and then solved by the perturb solution with the help of Poincare‐Lighthill perturbation methods. We also calculate the Sherwood number, the Nusselt number, and the skin friction, all of which are of relevance to engineers. We also examine how changing these variables affects the velocity distributions of skin friction, viscoelastic fluid, and dust particles. We look at how uniform suction/injection affects the speed and temperature distributions. It is important to note that even in the resonance instance, suction/ injection directs the boundary layer to develop unexpectedly.